Maybe try reaching out to counsel listed at the bottom of the link and ask your question.
https://www.arentfox.com/perspectives/alerts/biden-eo-federal-contractor-covid-19-safety-leaves-vaccination-mandates-and
The very short answer is that the EO pertaining to federal contractors is not a blanket requirement applicable to all current contractors. It is mandating that a vaccine clause be inserted into all new contracts. So, the number of contractors impacted by this right now - and so with standing to sue - is very small. Probably take a bit to get lawsuits filed.